U the gravitational force is proportional to 1/ R³ , the square of time period of revolution is proportional to 

$\mathrm{F}=\mathrm{k}/{\mathrm{R}}^3$
Now $\frac{{\mathrm{mv}}^2}{\mathrm{R}}=\frac{\mathrm{k}}{{\mathrm{R}}^3}\text{ or }\mathrm{V}=\frac{1}{\mathrm{R}}\sqrt{\left(\frac{\mathrm{k}}{\mathrm{m}}\right)}$
The frequency of revolution v is given by
$\mathrm{v}=\frac{\mathrm{V}}{2\mathrm{\pi R}}$
So, the time period of revolution
$\mathrm{T}=\frac{1}{\mathrm{v}}=\frac{2\mathrm{\pi R}}{V}=2{\mathrm{\pi R}}^2\sqrt{\left(\frac{\mathrm{m}}{\mathrm{k}}\right)}$
or $\mathrm{T}\propto {\mathrm{R}}^2\text{ or }{\mathrm{T}}^2\propto {\mathrm{R}}^4$